Whakaoti i te {l}{-4x+3y=13}{15x+3y=-6}

$$\left\{ \begin{array} { l } { - 4 x + 3 y = 13 } \\ { 15 x + 3 y = - 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$-4x+3y=13,15x+3y=-6$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$-4x+3y=13$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$-4x=-3y+13$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(-3y+13\right)$$

Whakareatia -\frac{1}{4} ki te -3y+13.

$$x=\frac{3}{4}y-\frac{13}{4}$$

Whakakapia te \frac{3y-13}{4} mō te x ki tērā atu whārite, 15x+3y=-6.

$$15\left(\frac{3}{4}y-\frac{13}{4}\right)+3y=-6$$

Whakareatia 15 ki te \frac{3y-13}{4}.

$$\frac{45}{4}y-\frac{195}{4}+3y=-6$$

Tāpiri \frac{45y}{4} ki te 3y.

$$\frac{57}{4}y-\frac{195}{4}=-6$$

Me tāpiri \frac{195}{4} ki ngā taha e rua o te whārite.

$$\frac{57}{4}y=\frac{171}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{57}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{3}{4}y-\frac{13}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{4}\times 3-\frac{13}{4}$$

Whakareatia \frac{3}{4} ki te 3.

$$x=\frac{9-13}{4}$$

Tāpiri -\frac{13}{4} ki te \frac{9}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-4x+3y=13,15x+3y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&3\\15&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&3\\15&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&3\\15&3\end{matrix}\right))\left(\begin{matrix}-4&3\\15&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\15&3\end{matrix}\right))\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\15&3\end{matrix}\right))\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\15&3\end{matrix}\right))\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-4\times 3-3\times 15}&-\frac{3}{-4\times 3-3\times 15}\\-\frac{15}{-4\times 3-3\times 15}&-\frac{4}{-4\times 3-3\times 15}\end{matrix}\right)\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}&\frac{1}{19}\\\frac{5}{19}&\frac{4}{57}\end{matrix}\right)\left(\begin{matrix}13\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{19}\times 13+\frac{1}{19}\left(-6\right)\\\frac{5}{19}\times 13+\frac{4}{57}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$-4x+3y=13,15x+3y=-6$$

Me tango 15x+3y=-6 mai i -4x+3y=13 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x-15x+3y-3y=13+6$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4x-15x=13+6$$

Tāpiri -4x ki te -15x.

$$-19x=13+6$$

Tāpiri 13 ki te 6.

$$-19x=19$$

Whakawehea ngā taha e rua ki te -19.

$$x=-1$$

Whakaurua te -1 mō x ki 15x+3y=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$15\left(-1\right)+3y=-6$$

Whakareatia 15 ki te -1.

$$-15+3y=-6$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$3y=9$$

Whakawehea ngā taha e rua ki te 3.

$$y=3$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=3$$